An invariance principle in k-dimensional extended renewal theory

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ ) v, τ= 1,· ··, k is the covariance matrix of In this paper the weak convergence of Z n in (D[0, ∞)) k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t 1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ 1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

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The exact order of normal approximation in bivariate renewal theory

Advances in Applied Probability ◽

10.1017/s0001867800035825 ◽

1981 ◽

Vol 13 (01) ◽

pp. 113-128 ◽

Cited By ~ 5

Author(s):

Ibrahim A. Ahmad

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Normal Approximation ◽

Renewal Theory ◽

Rates Of Convergence ◽

Partial Sums ◽

Exact Order ◽

First Passage ◽

The Central Limit Theorem ◽

Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

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The exact order of normal approximation in bivariate renewal theory

Advances in Applied Probability ◽

10.2307/1426470 ◽

1981 ◽

Vol 13 (1) ◽

pp. 113-128 ◽

Cited By ~ 4

Author(s):

Ibrahim A. Ahmad

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Normal Approximation ◽

Renewal Theory ◽

Rates Of Convergence ◽

Partial Sums ◽

Exact Order ◽

First Passage ◽

The Central Limit Theorem ◽

Passage Times

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.

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First-passage percolation processes with finite height

Journal of Applied Probability ◽

10.2307/3213944 ◽

1985 ◽

Vol 22 (4) ◽

pp. 766-775

Author(s):

Norbert Herrndorf

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

First Passage Percolation ◽

Horizontal Strip ◽

First Passage ◽

Time Constants ◽

Finite Height ◽

Special Cases ◽

Percolation Processes ◽

Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

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Limit theorems for first-passage times in linear and non-linear renewal theory

Advances in Applied Probability ◽

10.1017/s000186780002293x ◽

1984 ◽

Vol 16 (04) ◽

pp. 766-803 ◽

Cited By ~ 6

Author(s):

S. P. Lalley

Keyword(s):

Limit Theorems ◽

Large Deviation ◽

Local Limit Theorem ◽

Renewal Theory ◽

Local Limit ◽

Linear Boundary ◽

First Passage ◽

Non Linear ◽

First Time ◽

Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

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On the exact order of normal approximation in multivariate renewal theory

Journal of Applied Probability ◽

10.1017/s002190020003775x ◽

1985 ◽

Vol 22 (02) ◽

pp. 280-287 ◽

Cited By ~ 2

Author(s):

Ştefan P. Niculescu ◽

Edward Omey

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Normal Approximation ◽

Renewal Theory ◽

Rates Of Convergence ◽

Partial Sums ◽

Exact Order ◽

First Passage ◽

The Central Limit Theorem

Equivalence of rates of convergence in the central limit theorem for the vector of maximum sums and the corresponding first-passage variables is established. A similar result for the vector of partial sums and the corresponding renewal variables is also given. The results extend to several dimensions the bivariate results of Ahmad (1981).

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Limit theorems for first-passage times in linear and non-linear renewal theory

Advances in Applied Probability ◽

10.2307/1427340 ◽

1984 ◽

Vol 16 (4) ◽

pp. 766-803 ◽

Cited By ~ 31

Author(s):

S. P. Lalley

Keyword(s):

Limit Theorems ◽

Large Deviation ◽

Local Limit Theorem ◽

Renewal Theory ◽

Local Limit ◽

Linear Boundary ◽

First Passage ◽

Non Linear ◽

First Time ◽

Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

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First Passage Time Distribution of a Two-Dimensional Wiener Process with Drift

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003120 ◽

1993 ◽

Vol 7 (4) ◽

pp. 545-555 ◽

Cited By ~ 9

Author(s):

Marco Dominé ◽

Volkmar Pieper

Keyword(s):

Wiener Process ◽

First Passage Time ◽

Exit Time ◽

Passage Time ◽

Absorbing Boundary Conditions ◽

Two Dimensional ◽

First Exit Time ◽

First Passage ◽

Absorbing Boundary ◽

Starting Point

The two-dimensional correlated Wiener process (or Brownian motion) with drift is considered. The Fokker-Planck (or Kolmogorov forward) equation for the Wiener process (X1(t), X2(t)) is solved under absorbing boundary conditions on the lines x1= h1 and x2 = h2 and a fixed starting point (x0,1, x0,2). The first passage (or first exit) time when the process leaves the domain G = ( −∞, h1) × ( −∞, h2) is derived.

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Limit theorem for ﬁrst passage times in the random walk described by the generalization of the autoregressive process

Uzbek Mathematical Journal ◽

10.29229/uzmj.2020-4-11 ◽

2020 ◽

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Autoregressive Process ◽

First Passage Times ◽

First Passage ◽

Passage Times

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First-passage time of Markov processes to moving barriers

Journal of Applied Probability ◽

10.2307/3213688 ◽

1984 ◽

Vol 21 (4) ◽

pp. 695-709 ◽

Cited By ~ 42

Author(s):

Henry C. Tuckwell ◽

Frederic Y. M. Wan

Keyword(s):

Differential Equation ◽

Markov Processes ◽

First Passage Time ◽

Exit Time ◽

Passage Time ◽

First Exit Time ◽

First Passage ◽

Sample Paths ◽

First Exit Times ◽

First Passage Time Problems

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

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